Integral representation of $\psi^{(\nu)}(z)$ for $\nu>0$ but $\nu\not\in\mathbb{N}$

Question

I need the integral representation of the polygamma function for $n>0$ but $n\not\in\mathbb{N}$.

I searched both among Wolfram functions and Digital Library of Mathematical Functions

1. On Wolfram I found the representation of $\psi^{(n+\frac{1}{2})}(z)$ with a mixture of summation and series.
2. I also found on Wolfram some integral representations, but trying to generate them gives me non-convergent integral (those involving the regularized gamma function $Q(a,0,z)$)
3. I also found one in which it is defined through a boundary integral but I don't know how to generate it.

In the 2nd point (the one with $Q(a,0,z)$) although "On the real axis " is written in the path, the problem is that the integral is done on complex values:

3° representation: since $t\in(0,1)\Rightarrow\ln(t)<0\Rightarrow \ln(t)^{\nu}\in\mathbb{C}$ (since $\nu>0$ and $\nu\not\in\mathbb{N}$)
4° representation: since $t\in(0,\infty)\Rightarrow -\ln(t+1)<0\Rightarrow (-\ln(t+1))^{\nu}\in\mathbb{C}$ (since $\nu>0$ and $\nu\not\in\mathbb{N}$)
5° representation: since $t\in(0,\infty)\Rightarrow -t<0 \Rightarrow (-t)^{\nu}\in\mathbb{C}$ (since $\nu>0$ and $\nu\not\in\mathbb{N}$)

I would need a representation that doesn't involve complex numbers though, i.e. something like this: $$\psi^{(\nu)}(x)=\int_{a}^{b} f(\nu,t,x)\mathrm{d}t\qquad \text{where }(a,b)\subseteq\text{Domain}(f)$$

Can someone point me to some other site with useful formulas by chance?